UNIVERSITY OF CALGARY

Application of the Stefan Problem to the Modelling the Decomposition of a Gas Hydrate

Pipeline Plug

by

Emmanuel Christian Bentum

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE

DEGREE OF MASTER OF SCIENCE

GRADUATE PROGRAM IN CHEMICAL AND PETROLEUM ENGINEERING

CALGARY, ALBERTA

MAY, 2017

© Emmanuel Christian Bentum 2017

i

ABSTRACT

This study deals with the application of the Stefan problem to modelling dissociation of hydrate

plug in which the hydrates were formed from a gas mixture. In the previous attempt to simulate

the decomposition of a hydrate pipeline plug, the hydrates have always been assumed to be pure

methane, will lead to erroneous prediction for the rate of decomposition of a hydrate plug

because the presence of even a small amount of ethane and or propane could drastically alter the

three-phase equlibrium conditions for a gas hydrate formation.

In the current study, the Stefan problem for heat conduction at a moving boundary is written in

radial coordinates for the case of double sided-depressurization of a pipeline hydrate plug. The

plug is assumed to have formed in the presence of various mixtures of methane and ethane, some

of which formed structure I hydrates and some which formed structure II hydrates. The effect of

the gas mixture composition, on the rate of hydrate plug decomposition is included by

incorporating Giraldo and Clarke’s model, for the rate of decomposition of gas hydrates formed

from a gas mixture, into one of the boundary conditions. In formulating the equations, it was

assumed that the depressurization is always occurring at a pressure whose corresponding

equilibrium temperature is greater than 273.15K, thus, there was no need to include an ice phase.

The resulting partial differntial equation is highly non-linear and was solved by using the method

of lines. At the time of writing,there were no publication available in which the method of lines

had been applied to stefan problem in radial coordinates. A base case model was run, in which

only heat conduction was considered.

ii

The base case scenario was able to satisfactorily model the experimental data from Peters et al.

[1] with an Absolute Average deiviation (AAD) of 5%. The kinetic model was subsequently

applied to the base case scenario and and it was found that the results were almost identical to

those obtained without the kinetic term. From this, it was concluded that heat transfer controls

the decomposition of the methane hydrate plug at the base case conditions.

Subsequently, the model was used to simulate the decomposition of hydrate plugs formed from

the mixtures of methane and ethane, some of which formed were sI hydrates and some formed

were sII hydrates. Without the addition of the kinetic term there would be no means for including

the composition of the gas mixture, in the heat transfer equations.

A sensitivity analysis on the kinetic model was conducted. The geometric parameter which is

related to the surface area was investigated and it was found out that by changing the ratio from 1

to 4 times varied very little indicating that the parameter was not very sensitive to the kinetic

model. It was also observed that at pressures of 7.4MPa the rate of dissociation was heat

controlled. However, when the pressure was lowered to 3.4MPa intrinsic kinetics became more

predominant indicating a sharp difference between the heat transfer model and the kinetic model.

At a temperature of 273.15K the model showed that the dissociation rate was only heat transfer

controlled at a pressure of 7.8MPa. The Absolute Average Deviation (AAD) was less than 1%.

However as the temperature was increased to 275.15K and eventually to 277.15K there was a

sharp deviation from the kinetic model to the heat transfer model.

iii

ACKNOWLEDGEMENTS

The author would like to express his honest gratitude to his supervisor Dr. Matthew A.

Clarke for his patience, continuous support, encouragement and supervision of this thesis.

The author also expresses his appreciation to the members of the examining committee

for their valuable comments.

The author also thanks Dr. Amitabha Majumdar for his assistance .The author also wants

to mention his research group members: Marlon Mendoza and Fahd Alquatani for the excellent

support.

Dedication

This thesis is dedicated to my family.

iv

LIST OF FIGURES

Figure 1: structure I hydrate cavities Reproduced from Sloan and Koh [10] ................................. 3

Figure 2: structure II hydrate cavities Reproduced from Sloan and Koh [10] ................................ 3

Figure 3: (a) unit cell of structure I (b) unit cell of structure II Reproduced from Sloan and Koh

[10] .................................................................................................................................................. 4

Figure 4: Unit cell of structure H Reproduced from Sloan and Koh [10] ...................................... 5

Figure 5: Phase diagram for a water-gas-hydrate-gas system Reproduced from Giovanni and

Hester [4] ........................................................................................................................................ 8

Figure 6: Natural gas gravity chart Reproduced from Sloan and Koh [10] .................................... 9

Figure 7: Plug formation through aggregation in an oil-dominated system. Reproduced

from Sloan and Koh [10, p. 653] ................................................................................................ 12

Figure 8: Removal of hydrates by pigging method (Reproduced from Giovanni and Hester) [8] 13

Figure 9: Hydrate plug formed from inside a pipeline in Brazil (photo by Petrobas reproduced

from Koh et al.) [22] ..................................................................................................................... 13

Figure 10 Pipeline rupture owing to excessive pressure buildup generated by hydrate dissociation

(Reproduced from Giovanni and Hester [8] ................................................................................. 14

Figure 11: Hydrate plug dissociation by one sided depressurization. (Reproduced from Giovanni

and Hester [8] ................................................................................................................................ 14

Figure 12: A phase diagram showing the general hydrate dissociation methods (Reproduced from

Sloan and Koh. [10, p. 585] .......................................................................................................... 15

Figure 13: Radial dissociation of hydrate plug Reproduced from Peters et al. [1] ....................... 34

Figure 14: Hydrate plug dissociation schematic Reproduced from Hong et al. [68, p. 1851] ..... 35

v

Figure 15: Discretization scheme with fictitious enthalpy values, Reproduced from Chun et al

[71] ................................................................................................................................................ 42

Figure 16: Schematic Linear enthalpy distribution for one-dimensional grid system, Reproduced

from Chun et al [71] ...................................................................................................................... 42

Figure 17: Discretization Scheme at the interface, Reproduced from Lacoa et al. [66] ............... 45

Figure 18: Plot of temperature profiles at different radial positions ............................................. 51

Figure 19: Plot of Temperature profiles of both pure methane (C1) and mixture of

methane/ethane (C1/C2) hydrate composition at different radial positions ................................. 52

Figure 20:3 D Plot of Temperature Profile of methane/ethane mixture at different radial positions

....................................................................................................................................................... 53

Figure 21: 3 D Plot of Temperature Profile of pure methane at different radial positions ........... 54

Figure 22: Plot of Temperature Profiles with pure methane and mixture at different radial

positions using kinetic model........................................................................................................ 55

Figure 23: 3D plot of pure methane at different radial position using the kinetic model ............. 56

Figure 24: Plot of structure I hydrate dissociation rate compared with the experimental data [77]

....................................................................................................................................................... 61

Figure 25: Plot of dissociation rate heat model for structure I with experimental data of Peters et

al. [77] ........................................................................................................................................... 62

Figure 26: Plot of heat transfer model (HT) compared with kinetic model at different pressures.

at a geometric ratio of one ............................................................................................................ 64

Figure 27: Plot of heat transfer (HT) model with present model (Kinetic) at different

temperatures .................................................................................................................................. 66

Figure 28: Plot of hydrate kinetics using different geometric parameters .................................... 67

vi

LIST OF TABLES

Table 1: Geometry of cages [10] .................................................................................................... 2

Table 2: Jeffrey’s List of series of seven Hydrate Crystal structures [15] ...................................... 7

Table 3: Table showing the number of iterations for the method of lines using different ODE

solvers ........................................................................................................................................... 50

Table 4: Table of calculated stoichiometric values, W of hydrate mixtures (CH4+C2H6) ........... 59

Table 5: Simulation Parameters of the hydrate plug ..................................................................... 59

Table 6: Table of activation and intrinsic kinetics constants for methane and ethane hydrates

source [42] .................................................................................................................................... 60

Table 7: Simulation Parameters source [27] ................................................................................. 70

Table 8: Hydrate dissociation models, Reproduced from Sloan and Koh [10] ............................ 71

vii

TABLE OF CONTENTS

ABSTRACT ......................................................................................................................... i ACKNOWLEDGEMENTS ............................................................................................... iii

LIST OF FIGURES ........................................................................................................... iv LIST OF TABLES ............................................................................................................. vi TABLE OF CONTENTS .................................................................................................. vii LIST OF SYMBOLS ......................................................................................................... ix CHAPTER 1: INTRODUCTION ........................................................................................1

1.1 Historical Background .......................................................................................................... 1

1.2 Structures of Gas Hydrates ................................................................................................... 2

Structure I................................................................................................................................ 3

Structure II .............................................................................................................................. 4

Structure H .............................................................................................................................. 5

1.3 Phase Equilibrium for gas hydrates ...................................................................................... 8

1.4 Determination of equilibrium conditions of gas hydrates..................................................... 9

1.5 Hydrate as a potential of source of energy.......................................................................... 10

1.6 Formation and removal of hydrate plugs ............................................................................ 11

1.6.1 Hydrate forming conditions ......................................................................................... 11

1.6.2 Remediation and removal of Hydrate plug .................................................................. 13

1.7 REVIEW OF HYDRATE KINETIC MODELS .........................................................17 1.7.1 Review of hydrate dissociation models ....................................................................... 18

1.7.2 Review of hydrate plug decomposition kinetics .......................................................... 21

1.7 3 The effect of Kinetics on the dissociation rate of the hydrate plug ............................. 23

1.7.4 Review of solution techniques for moving boundary heat transfer problems. ............ 24

viii

1.7.5 Application of Method of lines in solving partial differential equation ...................... 29

1.7.6 Comparison between finite difference method and method of lines ........................... 31

1.8 Scope of study ..................................................................................................................... 32

CHAPTER 2 MODELLING OF HYDRATE PLUG DISSOCIATION RATE ...............34 2.1 Hydrate dissociation by double-sided depressurization ...................................................... 34

2.2 Derivation of the moving boundary equation with the Kinetic term .................................. 38

2.3 Derivation of moving boundary of the heat transfer model using method of lines in radial

coordinates ................................................................................................................................ 41

2.4 Derivation of the heat equation by method of lines in the cylindrical coordinates ............ 43

CHAPTER 3-RESULTS AND DISCUSSION .................................................................50 3.1: Simulation Results for Temperature Profile in the Pipeline .............................................. 50

3.2 Decomposition of Hydrate plug kinetics ............................................................................ 57

3.3 Sensitivity analysis of the Model ........................................................................................ 64

CHAPTER 4: CONCLUSIONS AND RECOMMENDATIONS .....................................68

4.1 Conclusions ......................................................................................................................... 68

4.2 Recommendations ............................................................................................................... 69

APPENDIX ........................................................................................................................70 BIBLIOGRAPHY ..............................................................................................................72

ix

LIST OF SYMBOLS

Ap surface area of hydrate, m2

C Langmuir constant, 1/Pa

Cbo initial concentration, mole/dm

3

Ceq equilibrium concentration of dissolved gas in the presence of hydrate, mole/dm3

Cint concentration, mole/dm3

CpI heat capacity of ice phase, J/ (Kg. K)

Cpw heat capacity of water phase, J/ (Kg. K)

dn change in moles

dt change in time, s

e exponent

Ea Activation energy, KJ/mol

f fugacity, MPa

feq equilibrium fugacity, MPa

G linear dissociation rate, m/s

K* kinetic constant, Overall rate constant around a hydrate particle, mol/(m2.Mpa.s)

Kb mass transfer coefficient from liquid bulk to the surface of the particle, m/s

KI thermal conductivity of ice, watts per meter Kelvin, W/(m.K)

Kw thermal conductivity of water, watts per meter Kelvin, W/(m.K)

Ms molecular weight g/mol

n(t) number of moles remaining in the hydrate and water at the start of dissociation, moles

Nw Number of water moles

x

P Pressure, MPa

R Universal gas constant, 8.314KJ/ (mole. K)

R radial distance in pipe, cm

ro radial position at pipewall ,cm

Ry(t) Global rate of reactions, mole/(m3.s)

s1 water-hydrate interface position layer, cm

Td dissociation Temperature, K

Teq equilibrium Temperature, K

Tm melting Temperature, K

Tw Temperature water in pipe, K

V volume of mass reaction, m3

W stoichiometric amount of moles with respect to methane

W(r) cell potential function, J

xb gas mole fraction in the liquid bulk in equilibrium with hydrates phase

xint gas mole fraction in the liquid bulk in equilibrium with gas phase at the interface

yi vapour phase mole fraction of species ‘i’

yi,j Fractional occupancy of cavity species ‘ i’ in cavity j

Z compressibility factor

Ψ geometric ratio of hydrate plug

фs Surface volume of hydrate plug, cm3

фv Volume of hydrate plug, cm3

∆r radial grid size, cm

xi

Greek Letters

heat of dissociation of hydrate, J/(Kg. K)

heat of dissociation of ice, J/(Kg. K)

∆R radial change in position, cm

∆t time step, s

Pi

Density, kg/m3

Porosity

Thermal diffusivity of water, m2/s

Ψ Sphericity

νi number of cavities per type i

. Partial fugacity coefficient,

xii

Superscripts

H hydrate

MT empty lattice

o pure liquid water at reference conditions

Subscripts

eq equilibrium condition

s system condition

i index

j index

o reference conditions

w water

m molecule

b bulk solution

p particle

g gas

1

CHAPTER 1: INTRODUCTION

1.1 Historical Background

Gas hydrates are non-stoichiometric crystalline compounds which are formed as result of

association of water molecules and lower molecular weight mostly non polar gases under low

temperature and elevated pressures. The water forms three dimensional network structures with

spaces that can be occupied by these low molecular gases like methane, ethane and carbon

dioxide. The forces holding the hydrates together are the weak van der Waal forces. In 1778 Sir

Joseph Priestley [2] first formed hydrates from ice and sulphur during the winter season.

However, it was in 1810 that Davy [3] discovered that hydrates could be formed by reacting

chlorine with water. Faraday in 1823 [4] came out with the formula for chlorine hydrates. Then,

in 1888 it was discovered that hydrates could be formed with the gaseous hydrocarbons like

methane, ethane and propane. [5]

During 1930 Hammerschmidt [6] found out that the blockages in oil and gas pipelines were due

to the presence of hydrates. It soon become known that hydrate pose serious threat to petroleum

productions and explorations. At these operating conditions of high pressure and low

temperature the hydrates formed could block other flowlines like, subsea lines, risers, blow-out

preventer (BOP) and chokelines.

The hydrate plug formed could lead to massive economic losses, destruction to life and property

in the oil and gas industry if best practices and safety measures are not put in place. In 1967 [7],

a huge deposit of naturally occurring hydrates were found under the permafrost regions of

2

Siberia. Later on more methane hydrate were discovered in the Alaska, Mackenzie delta in

Canada [8]. Though it has been found to be a nuisance to the oil industry it been seen to be

potential store of energy [8]. The potential amount of natural gas stored in situ gas hydrate in the

world has been estimated to be twice the amount of the world’s conventional gas reserves [9].

Ever since its discovery increased research has been conducted to better understand how to

control formation of hydrates and more recently it kinetics.

1.2 Structures of Gas Hydrates

There are three main types of hydrates structures namely Structure I, Structure II and Structure

H. The determination of the structures sI and sII were done using X-ray diffraction techniques

[10] Table 1 shown below gives the structural formula of the types of hydrates. The structural

formulae are illustrated in the Figures 2, 3 and 4.

Table 1: Geometry of cages [10]

Hydrate crystal

structure

I II H

Cavity Small Large Small Large Small Medium Large

Description 5

12 5

126

2 5

12 5

12 5

12 4

35

66

3 5

126

8

Number of cavities 2 6 16 8 3 2 1

Average cavity

radius (Å)

3.95 4.33 3.91 4.73 3.94 4.04 5.79

No. of water

molecules/cavity

20 24 20 28 20 20 36

3

Figure 1: structure I hydrate cavities Reproduced from Sloan and Koh [10]

Structure I

The unit cell of structure I is pentagonal dodecahedra which are packed together with

tetrakaidecahedra to form twelve pentagonal and two hexagonal faces. The sI is made up of 46

water molecules, two small 512

cavities and two large cavities 512

62. The structure I can be

occupied with small sized guest molecules which is less than 3 Å in molecular radius such as

methane, ethane, carbon dioxide, and hydrogen sulfide [10]

Figure 2: structure II hydrate cavities Reproduced from Sloan and Koh [10]

4

Structure II

The unit cell of structure II is also pentagonal dodecahedra which are packed together with

hexakaidecahedra to form twelve pentagonal and four hexagonal faces. Structure II is made up of

146 water molecules, sixteen small 512

cavities and eight large cavities 512

64. This structure can

be occupied by both small and larger sized molecules; for instance, propane and isobutene may

be entrapped in the larger cavities.

Figure 3: (a) unit cell of structure I (b) unit cell of structure II Reproduced from Sloan and Koh [10]

5

Structure H

Structure H unit cell is made up of 34 water molecules, three small 512

cavities, two medium size

435

66

3 cavities and one large size 5

126

8 cavity. The small guest molecules usually are caged in

small and medium cavities whereas molecules larger than 7.4Å, such as 2-methylbutane, 2, 2-

methylbutane, neohexane and cyclo-heptane, enter the larger cavity.

Figure 4: Unit cell of structure H Reproduced from Sloan and Koh [10]

The three main cavities in sH gas hydrates are the pentagonal dodecahedron 512

,

tetrakaidecahedra 512

62, hexakaidecahedron 5

126

4, irregular dodecahedron 4

35

66

3 and

icosahedrons (512

68). The 14 sided cavity tetrakaidecahedron is also referred as 5

126

2 since it has

12 pentagonal and 2 hexagonal faces.

The 16-hedron (hexakaidecahedrai cavity) are represented by the formula 512

64 because in

addition to 12 pentagonal faces it contains 4 hexagonal faces. The irregular dodecahedron cavity

(435

66

3) consists of three square faces and six pentagonal faces together with three hexagonal

faces. The biggest icosahedron cavity (512

68) has 12 pentagonal faces together with a girdle of 6-

6

hexagonal faces and a hexagonal face each at the cavity crown and foot. Several other forms of

hydrates have been discovered by some researchers. In the analysis of the simple and combined

cavities of Dyandin et al. [11], it was proposed that in addition to the cavities present in sI, sII

and sH were the following structures 512

63,4

45

4, 4

35

96

27

3and 4

66

8. The cavities expand in size in

comparison to ice and it is stabilized by the repulsive presence of the hydrate formers (guest

molecules) or the neighbouring molecules.

Tabushi et al. [12] also found out that the 15-hedron (512

63) was absent in clathrate except

bromine owing to an unfavourable strain in relation to the other cavities in structures I and II.

Rogers and Yevi [13] indicated that guest repulsion is more pronounced than attraction which

leads to cavity expansion. The mean polyhedral volume 12, 14 and 16 -hedral cavities were

found to vary with temperature, guest size and shape as observed by Chakumakos et al. [14].

However, Jeffery [15] as shown in Table 2 found out that 12, 14 and 16 -hedral cavities are not

stable in pure water structure. Sorensen and Walrafen [16] found out that liquid water may be

structured as cavities.

7

Table 2: Jeffrey’s List of series of seven Hydrate Crystal structures [15]

1 11 111 IV V VI VII

12-Hedra

512

12-Hedra

512

12-Hedra

512

12-Hedra

512

12-Hedra

512

8-Hedra

445

4

14-Hedra

512

62

16-Hedra

512

64

14-Hedra

512

62

15Hedra

(512

63)

14Hedra

(512

62)

512

63

16-Hedra

(512

64)

17-Hedra

(435

96

27

3)

14-Hedra

(466

8)

Cubic(s1)

Cubic(s11) Tetragonal Hexagonal Hexagonal Cubic Cubic

Pm3n Fd3m P42/mnm P6/mmm P63/mmc 143d Im3m

a=12Ả a=17.3 Ả c 12.4 Ả

a 25.5 Ả

c Ả

a

c ,

a 12 Ả

a=18.8(2) A=7.7 Ả

6x.2Y.

46.H 20

8X.16Y.

136.H 20

20X.10Y.

172.H20

8X.6Y.

80.H20

4X.8Y.

68.H2 0

16X156H20 2X.12H20

Gas

hydrates

Gas

hydrates

Bromine

hydrate

None

known

None

known

Me3CNH2

hydrate

HPF6

hydrate

X indicates the guest molecules in 14-hedra or larger voids, Y refers to those in 12-hedra

8

1.3 Phase Equilibrium for gas hydrates

The phase equilibria of gas hydrates provide the most significant set of properties that determine

the boundary for which hydrate exists. Figure 5 illustrates the key features of a phase diagram

when hydrates form from pure hydrocarbons.

Lw-H-G is the liquid water-gas hydrate-gas equilibrium line; I-H-G is the ice-gas hydrate-gas

equilibrium line and therefore at F if at constant temperature a hydrate plugs dissociation occurs

by moving F through equilibrium point E and finally to point D through depressurization.

Figure 5: Phase diagram for a water-gas-hydrate-gas system Reproduced

from Giovanni and Hester [4]

9

1.4 Determination of equilibrium conditions of gas hydrates

The equilibrium conditions of gas hydrates can be determined by the as gravity method. Using

the criteria for phase equilibrium the following conditions must be met.

1. Temperature and pressure of all the phase in the equilibria must be equal

2. Chemical potential of each component in each phase must be equal

3. Gibbs free energy is minimum

The gas gravity method was introduced by Katz [17] and it is a simple graphical technique in

which the gravity of the gas is required for determination of the equilibrium conditions. Gas

gravity is defined as the ratio of the molecular weight of the gas over the density of air. After

obtaining either temperature or pressure and the gas gravity, the other variable (either pressure or

temperature) can be read directly from the graph in Figure 6

Figure 6: Natural gas gravity chart Reproduced from Sloan and Koh [10]

10

1.5 Hydrate as a potential of source of energy

Gas hydrates can be potential source of energy. They are present in huge quantities in all the

continental shelves and in permafrost areas. The technology required for the utilization of this

energy is currently under research. Understanding the geology and properties of the reservoir is

very important in the successful utilization of this resource. Current estimates by the National

Research Council [18] and also by Association of America Petroleum engineers and Geologist

AAPG [19] estimates that the in-place volume of gas in gas hydrates for Gulf of Mexico to be

about 600TCM (6x1014

m3) and at North Slope of Alaska with a mean estimate of 2.4TCM

(2.4x1012

m3).

In 2009 China made a huge discovery of onshore deposit at Qinghai province and Tibet plateau

at a depth of 130-300m below the permafrost [8]. Economic and commercial production of

hydrates is being considered by Japan and other nations like USA, South Korea and Canada in

the coming decades [8]. Japan drilled an offshore methane hydrate production test well in 2013.

This research will convert natural gas to gas hydrates for easier transportability and avoid the

prohibitive cost of pipeline transport.

11

1.6 Formation and removal of hydrate plugs

Hydrate plugs are may form when there is a change in flow geometry (e.g. bend, or pipeline dip

along the sea floor). It may also occur at a nucleation site (e.g. sand, weld slag etc). Hydrate

plugs may also occur during transient operations for instance during an emergency shut-down

due to the failure of inhibitor injection failure or dehydrator failure. It may also occur following

restart after shutdown.

Another possibility of formation occurs when it is made to flow through a valve which may

result in further cooling. Structural imperfections, weld spot and pipeline fitting like elbow, tee

and valve are suitable sites for nucleation of hydrates. Also high velocity flow which occurs in

narrow orifices and valves causes mixing, which may enhance the formation of hydrates. When

natural gas flow through choke valves, hydrates may form is due to joule Thomson effect [20] .

1.6.1 Hydrate forming conditions

Formation of hydrate normally happens when conditions of elevated pressure and low

temperature are present in undersea pipelines. Other causes of hydrate formation are, natural gas

below its water dew point in the presence of water, turbulence and high velocity in the flowlines

as reported by Ikoku [21] . Again, the presence of H2S and CO2 which has a higher solubility

than hydrocarbons [21] could also promote hydrate formation.

12

Figure 7: Plug formation through aggregation in an oil-dominated system. Reproduced from Sloan and Koh [10, p. 653]

The Figure 7 above shows the process of agglomeration leading to the plug formation. The

process of plug formation begins at the nucleation site, then hydrate particles begins to grow with

time. Then the hydrate formed begins to aggregate leading to the plugging of the pipeline. There

has been steady progress in allowing hydrates to be formed but inhibiting the aggregation

process to avoid the formation of the hydrate plug. Hydrates are in transportable form and could

flow in flowlines as reported by Sloan and Koh [10]

13

1.6.2 Remediation and removal of Hydrate plug

When a hydrate plug is formed in a pipeline it needs to be remediated almost immediately to

mitigate economic losses. When hydrate forms there is down-time in production leading to

revenue shortfalls, and extra cost for remediation.

Figure 8: Removal of hydrates by pigging method (Reproduced from Giovanni and Hester) [8]

Figure 9: Hydrate plug formed from inside a pipeline in Brazil (photo by Petrobas reproduced from Koh et al.) [22]

If the hydrate formation is slow and is able to flow, pigging is the most suitable method

employed which can be used to clean and clear out the pipeline. The other methods of

remediation include injecting chemical inhibitors, heating, mechanical and depressurization

source [8].

14

One or a combination of methods can be employed. Each of the methods carries its own potential

dangers. When using thermal method care must be taken to avoid excessive pressure build-ups

and possible rupture and explosion as shown in Figure 10. The depressurization technique is the

most widely used in field operations where absolute hydrate plugs in pipelines are detected. One

sided depressurization is quite dangerous since the plug may be dislodged and move swiftly as a

projectile and cause equipment damage and injury to personnel as illustrated in Figure.10.

Figure 10 Pipeline rupture owing to excessive pressure buildup generated by hydrate dissociation (Reproduced from

Giovanni and Hester [8]

Figure 11: Hydrate plug dissociation by one sided depressurization. (Reproduced from Giovanni and Hester [8]

Lysne in 1995 thesis [23] reported three incidences in which hydrate projectiles exploded from

pipelines at elbows causing the death of three oil workers and loss of over US$7 million in

capital cost.

15

Figure 12: A phase diagram showing the general hydrate dissociation methods (Reproduced from Sloan and Koh. [10, p.

585]

As shown above in Figure 12 depressurization results in the heat being transferred to the hydrate

plug to decompose it. The heat is absorbed to decompose the hydrate at a constant interface

temperature. For the broken lines the inhibitor injection is represented by shifting the hydrate

formation curve by injection of 10 (w/v) % methanols in the free water phase.

In practice two sided- depressurization is the most recommended method over the single-sided

method. Double-sided depressurization method has become the preferred choice for removing

hydrate plugs in undersea natural gas and condensate pipelines as observed by Yousif and

Donayevsky [24].

16

Hydrate dissociation by depressurization is a heat and mass transfer dependent process. This

process involves a gradual reduction of pressure below the hydrate equilibrium pressure at which

the hydrates will be unstable as temperature rises at the hydrate interface. This method does not

necessarily require knowledge of its location, size or composition before it is implemented.

However, caution is required not to depressurize the pipeline quickly without heat transfer

(adiabatic) as Joule-Thomson cooling may rather worsen the problem by causing ice formation.

Again if the depressurization is done very slowly (isothermal depressurization) is also not the

most suitable option.

The most ideal approach is to use intermediate depressurization which ensures that the hydrate

interface temperature is always much lower than the surrounding temperature which will allow

the heat move from the surroundings to dissociate the hydrate from the pipe boundary inwardly.

In addition to finding suitable and economical methods of remediating hydrate plug, an offshore

pipeline will need suitable techniques of dealing with hydrate plugs formed in pipeline which are

submerged in very deep and cold waters especially if the plug is a few kilometers away from

shore. Formation of such a plug may potentially occur in unplanned shutdown probably due to

equipment failures. Hence thorough understanding of the mechanisms and processes involved is

key in developing techniques in controlling hydrates plugs.

17

1.7 REVIEW OF HYDRATE KINETIC MODELS

Clathrate hydrates have gained a very huge attention during these recent past decades owing to

its different potential application in its transportation, storage of natural gas and carbon

sequestration in the ocean. These applications require the development of effective hydrate

reactors thus requiring a deeper understanding of hydrate dissociation kinetics. Unlike hydrate

thermodynamics hydrate kinetics are still not fully understood.

In this section review of the literature on hydrate kinetics is presented which lays emphasis on

modeling. The models presented touches on the techniques of hydrate decomposition with its

kinetics investigated. The various dissociation models developed by most research groups

indicate that a greater number of them are based on heat transfer. Again some of the heat transfer

models have been coupled with kinetics in other to investigate the intrinsic kinetics at different

temperatures and pressures. Furthermore it has been observed that based on the comparisons

with experimental data most of the models are heat controlled rather than intrinsic kinetic

controlled [22].

Gupta et al. [25] in 2006 investigated the dissociation of methane hydrate by the use of nuclear

magnetic resonance and found out that intrinsic kinetics did not play a significant role in the

overall process. Hydrate dissociation is mainly controlled by heat transfer with kinetics playing

an insignificant role in majority of the cases.

\

18

1.7.1 Review of hydrate dissociation models

Kelkar et al. [26] used a model based on rectilinear coordinates that suggested that at an

optimum pressure the most rapid dissociation of the hydrates or solid phase occurs. However,

rapid depressurization caused ice to form due to joule Thomson cooling which delayed the

dissociation rate though ice had a higher thermal diffusivity which transmitted higher heat flux.

Thus, for low pressure depressurization the ice melts more slowly than the hydrate.

Jamaluddin et al. [27] investigated the decomposition of hydrate plugs and used a mathematical

model which coupled intrinsic kinetics with mass and heat transfer model to compare

experiments carried out under the condition of shutdown in pipeline in the laboratory at

temperature of 274K and pressures of 4MPa, 5MPa, and 7MPa.The model was used to compare

experimental values which indicated that if there was planned shut down under 48hours plugging

may not occur based on the effective diffusivities of methane gas through the hydrate. The

accurate estimation of the effective diffusivities is vital to the model prediction.

Vazquez-Roman [28] used a one -dimensional flow model to calculate pressure and temperature

profiles of gas production system based on average velocity between inlet and outlet conditions

of the pipe. The flow model was coupled with a heat transfer model to account for the

surrounding effects. The diameter, length and overall heat transfer coefficient of the hydrate plug

were then fine-tuned in the model to match the experimental data of the pressure and temperature

profiles which significantly improve the accuracy of its predictions.

19

Bollavaram et al. [29] investigated hydrate dissociation mechanism by method depressurization

for both (two-sided and single-sided). The model was based on Peters et al. [1] model that used

the two sided hydrate dissociation as radial moving boundary model. The model was able to

estimate the hydrate dissociation time and the total time for the complete melting of the hydrate

plug. The one-sided depressurization model was also investigated .This single sided model was

able to predict the time required to re-start the flow in a pipeline. This time also was dependent

on the downstream pressure, length, porosity, and permeability of the plug.

A safety model was developed to estimate a safe and optimum pressure for one sided

depressurization. The one-sided model was validated with laboratory and Tommeliten field plugs

and the predicted gas evolution curves matched the data [30] within 10% absolute error. The

model predicted that the Tommeliten field plugs were re-started when the annulus spacing was

8% of pipeline radius. The safety model also compared well with the simulations of Xiao et al.

[31] but predicted higher comparable plug velocities.

Nguyen et al. [32] developed a numerical model to predict the dissociation time of hydrate plugs

in oil subsea pipelines. The experimental hydrate plugs were dissociated by the method of

symmetric depressurization the model was in very good agreement with their experimental data.

The dissociation of the hydrate plug was dependent on the following variables diameter, porosity

and the dissociation temperature (Td).

20

Osokogwu and Ajienka [33] presented model that was based on Fourier heat law. The model

considered a radial dissociation from two-sided approach on a fixed boundary which applied

depressurization technique in a pipeline. Results from the model indicated at initial dissociation

temperature of 285.7K and a dissociation time was 13hours.

Vlasov [34] developed a model based on theory of chemical kinetics for the dissociation of gas

hydrate. The driving force for dissociation was highlighted in the model and the rate constant of

the hydrate dissociation was found to be dependent on the temperature. It was found out that the

temperature reliance of hydrate dissociation was confirmed with available experimental data in

the case where the interface involved liquid water. The case involving ice interface was yet to be

determined.

Uddin et al. [35] has proposed a new model based on deep investigation of a previous laboratory

scale study of methane hydrate decomposition and some observations from molecular dynamics

study. The model has not been tested with the appropriate data yet.

Lekvan et al. [36] investigated methane hydrate dissociation rate as a function of temperature.

Using a model that was based on the pseudo-elementary processes, it was able to predict the

kinetics using temperature gradients. The temperature range used for the study was narrow and

therefore was limited in its application in hydrates in subsea pipelines.

21

Chen et al. [37] used previous experimental data that were carried out in the laboratory for gases

in pipelines. A hydrodynamics model with an integrated model was used to simulate gas

dissociation. The model based on Englezos et al. [38] was used to estimate rate of evolution of

gas bubbles from the hydrates as it moved upward and also estimate the range of values of D

(mass transfer coefficient in the dispersion film) for which it was sensitive.

Rehder et al. [39] measured the dissociation rates of methane and carbon dioxide hydrates in

seawater underfloor. The hydrate dissociation was due to the differences between the

concentration of the guest molecule in the hydrate interface and the bulk solution. Thus a

solubility-controlled boundary layer model was able to satisfactorily predict the dissociation

data. The dissociation rate of carbon dioxide was higher than methane due to its higher solubility

in water.

1.7.2 Review of hydrate plug decomposition kinetics

In 1987, Kim et al. [40] studied the kinetics of methane hydrate dissociation. He found that the

dissociation rate was directly proportional to the surface area of the hydrate particle and the

fugacity difference of methane at equilibrium pressure and the decomposition pressure. An

experimental determination of the hydrate particle diameter led to the development of the

intrinsic model for the kinetics of hydrate decomposition.

22

Yousif et al. [24] simulated methane hydrate dissociation using a three-phase ID model .The

model matched very well the experimental data of gas and water produced from depressurization

and the movement of dissociation front .

Clarke and Bishnoi [41] developed a mathematical model to estimate the intrinsic rate constant

of decomposition. The model accounted for the particle size distribution in the hydrate phase as

the intrinsic rate constant was determined. Data was obtained from experiments performed using

temperatures of 274.15 to 281.15K and at pressures between 5 and 11bar was used to estimate

the intrinsic rate constant. The intrinsic rate constant was estimated to be 2.56.x103

mol/m2Pas.

Giraldo and Clarke [42] presented the experimental data from the kinetics of decomposition of

hydrates from pure ethane and methane and its mixtures 25%-75% methane at temperature and

pressure range of 274-278K and 6.39 and 14.88bar respectively. The model predicted accurately

well the experiment data for sI hydrates. For gas mixtures that formed sII hydrates

(72%CH4/28%C2H6 and 75%CH4/25%C2H6) the rate constants for both methane and ethane in

sII was regressed. The intrinsic rate constant of ethane decomposition was assumed to be the

same for sI and sII. The intrinsic rate constant regressed for methane in sI was estimated to be

8.06x103

mol/m2Pas

23

1.7 3 The effect of Kinetics on the dissociation rate of the hydrate plug

When kinetics is negligible or intrinsic kinetics is extremely fast (i.e. heat transfer only), the

interface fugacity becomes nearly equal to the equilibrium fugacity and therefore heat transfer

equation is employed to determine the dissociation rate of the hydrate plug. The hydrate plug is

assumed to attain extremely fast the interface temperature, Td. Even though the gas phase has a

much higher resistance than the liquid phase, it is generally assumed that all the heat absorbed by

the dissociating hydrate at the hydrate-water interface is conducted only through the water from

the pipewall.

In order to investigate if kinetics is influencing the dissociation rate (i.e. kinetics model) would

be applied for the simulation study. The driving force for the dissociation becomes the difference

in fugacity of methane/ethane mixture at the interface and the equilibrium fugacity of the system,

as reported by Englezos et al. [38] . The temperature distribution in the water phase which is

dependent on the temperature gradient between the pipewall and the interface and the radial

distance which could be determined by equations (17) and (18).

The hydrate plug is assumed to be a solid which occupy the region 0<R<s(t) which divides the

two phases(i.e. hydrae-water). The pipewall temperature is assumed to be at a constant

temperature, To and Tm which is the temperature at the interface is also assumed to be constant.

The surface of the hydrate plug is initially regarded as rough. As the hydrate plug dissociates

with time it surface becomes smoother with time at the latter stages.

24

Therefore the value of the dimensionless surface ratio , approaches the value of unity. The

intrinsic kinetic rate of hydrate dissociation is proportional to the surface area. The dimensionless

parameter (surface roughness factor), which indicates the real surface area ratio to the geometric

one is difficult to determine. Due to the difficulty in determining the geometric parameter Ψ,

however, the value of unity was assumed for it in the modeling by Englezos et al. [38] in their

experimental determination of the kinetic rate constant.

1.7.4 Review of solution techniques for moving boundary heat transfer problems.

When hydrate plugs begins to dissociate it involves heat transfer at the moving boundary.

Several mathematical and numerical techniques have been employed in solving this type of

problem which is usually referred as Stefan [43] type moving boundary involving phase changes

such as from solid from liquid.

Verma [44] et al. used the fixed grid method, which was based on explicit finite difference

approach to solve the moving boundary problem. This method used an alternative equation to

describe the grid containing the interface with the energy balance of the grid point next to the

interface. This equation reduced the mass balance error that generally affected the numerical

solution of moving boundary problems.

In addition to the increased accuracy of its prediction due to the elimination of the mass balance

error, the method had the advantages of a fully implicit scheme. The results given by that

approach was identical to the one developed by Murray and Landis [45] who employed fictitious

temperature which contained the fusion front. They used variable grid method by discretizing

either the time or space domain into equal sized increments, while the other is allowed to vary in

25

grid size. The time increment is maintained constant and the liquid and solid regions are at any

instant in time divided into a fixed number of equally spaced intervals. However, the size of the

intervals varied as the proportion of solid/liquid changes, this type of subdivision presented some

difficulties for the initial start up of the numerical computation particularly when the liquid or

solid is very small in dimension.

Kutluay [46] used variable space grid and boundary immobilization method which was based on

the explicit finite difference to determine temperature history and interface position. The variable

space grid which always ensures that the moving boundary is always on the Nth

grid is achieved

by the varying the grid size with time. However, the boundary immobilization method which was

first proposed by Laudau [47] is much more convenient is implemented by transforming the

moving boundary into a fixed boundary. This approached significantly simplifies the numerical

method and by applying the finite difference method a solution is obtain with high accuracy and

convergence is easily attained.

Chernousko [48] independently studied the isotherm migration method with Dix and Cizek [49].

In this method the dependent variable (Temperature) can be interchanged with the space

variable ( R) so that the solution is evaluated as R(T, t) instead of the more common T(R,t).This

method provides fixed boundaries in solving the model problem. Thus the method is most often

used to obtain numerical solutions especially subject to time dependent boundary conditions.

Esen and Kutluay [50] [went further to apply a Neumann boundary condition to obtain numerical

solutions of the Stefan problem.

26

Myers and Mitchell [51] presented the combined integral method (CIM) in which the Stefan

problem was solved by the introduction of a heat penetration depth. This was followed by an

approximating function in a polynomial form that described the temperature range .Then the heat

equation was integrated over that heat range to obtain heat balance integral. Then the resultant

single ordinary differential equation was solved analytically to determine the interface position.

Reutskiy [52] introduced a meshless technique (which did not need any domain or boundary

discretization) for solving one-dimensional problems for the moving boundary. The key idea of

this method was the use of modified particular solutions which satisfy the homogenous boundary

conditions. This technique utilizes truncated Fourier series as approximate fundamental solutions

The Fourier representation allows to write the solution on each time-layer, the right hand side

with the particular solution in the same form of truncated series resulting into two unknowns. It

was based on the application of delta-shaped functions and the method of approximate

fundamental solutions. The results show remarkable very high accuracy in tracking the interface

position. Again it was further produced higher accuracy results for cylindrical geometries and 2-

Dimensional Stefan problems where comparison was very good with analytical solutions by

Chein-Shan Liu [53].

Case and Tausch [54] used Green’s method to transform the partial differential equation into a

system of integral equations with unknown fluxes on the interface. This is discretized with

Nystrom method using the Stefan condition to track interface position. The results were quite

accurate.

27

Mitchel and Vynnycky [55] used numerical algorithm for one-dimensional time dependent

Stefan problem by Keller box finite difference scheme. The significance of the work was the use

of variable transformation that was built into the numerical algorithm to resolve the boundary

condition discontinuity arising from the onset of phase change. The method allowed the delay

time until dissociation/melting begins to be determined. Its results gave a second order accuracy

in both time and space.

.

Papac et al. [56] applied a numerical method using a hybrid finite-difference and finite-volume

framework which also encompassed the level-set finite difference discretization. The goal of this

approach (level-set finite difference) is to compute and predict the moving interface under

velocity field. This velocity depended on the time, position and the geometry of the interface.

This method is very simple and adaptable in its implementation.

Tadi [57] used a fixed grid local method by coordinate transformation which could be applied to

obtain exact solutions within a small local region. The method is used to cover the whole domain

resulting into implicit scheme with first order accurate in time. Ping et al. [58] presented Stefan

problem with exact solution method to solve the one-dimensional elliptic problem with Dirichlet

boundary condition. Results generated were quite accurate.

Voller [59] employed a sharp and diffuse interface model of fractional Stefan problems with

implicit time stepping numerical solution for the diffuse interface fractional Stefan model. The

results were accurate and the method was able to capture sharp physical changes

28

Hetmaniok et al. [60] employed the alternating phase truncation method to solve two

dimensional inverse Stefan problem based on the knowledge of selected locations of the region

of interest to determine the heat transfer coefficient. Each phase is treated alternately which

simplifies into algebraic problem. It is thus easy to implement and avoids numerical instability.

Song et el. [61] used isogeometric approach to solve the Stefan problem using algebraic distance

estimations and point projection algorithms developed under the condition of Gibbs-Thomson

conditions. Its method was much more effective than Newton-Raphson iterations.

Krasnova and Levashov [62] used numerical approach to solve one-dimensional two-phase

Stefan problem based on ultrafast processes in solids (which lasted less than the time of electron-

phonon relaxation) paving way for the heat equations to be solved separately for ions and

electrons at constant volume. The numerical method derived is used to determine ionic and

electronic temperatures in aluminum alloy subject to a femtosecond (10-15

s). The ionic

temperature showed a jump at the Stefan condition whereas the electronic temperature did not.

However the kinetics of the melting process had an effect on the model prediction.

Myers and Font [63] solved the one-phase Stefan problem by transformation of the phase change

temperature to a variable using, Cartesian, cylindrical and spherical coordinates. The resultant

equations were simple to implement very adaptable. Mitchell [64] applied the method of

combined integral method (CIM) which is characterized by delayed onset of phase change based

on Robin condition. The method was stable and versatile.

29

Layeni and Johnson [65] used a differential-difference equation reformulation exact closed-form

solution to a class of Stefan problems. The transformation reduces into analytical sets of

equations which is easily solved. The accuracy is high and very easy to implement. In this work

method of lines is employed by discretizing the partial differential equation in the space

dimension and leaving the time domain continuous as adopted by Campos and Lacoa [66] .

1.7.5 Application of Method of lines in solving partial differential equation

The method of lines is a well-developed numerical technique sometimes referred as a semi-

numerical method which was originally employed in the analysis of transmission lines wave

guide structures and scattering problems. It was initially developed by mathematicians and

applied to solve boundary value problems [67] .

It is a special numerical technique in which the space variable is discretized in one or two

dimensions while the time variable is allowed to be continuous or maintained in its analytical

form in a given differential equation. Thus this technique has both the merit of the finite

difference method and analytical method which does not produce spurious mode nor oscillatory

solutions and has no problem relating to relative convergence. Even though method of lines was

initially used to solve hyperbolic wave equation it could be applied to solve parabolic heat

equation. In this simulation study the method of lines would be applied to solve the partial

differential equation from Fourier’s heat equation.

30

The justification of using method of lines is its high computational efficiency. The presence of its

semi-analytical part makes it simple to make compact algorithm while at the same instant

yielding more accurate results. Again less computational effort is required as compared to the

finite difference numerical methods. Since the discretized space variable is analyzed separately

from the continuous time variable it is easy to achieve numerical stability and convergence for

wide range of problems.

Programming effort is also significantly reduced by employing the well-established and

dependable Ordinary differential Equation (ODE) solvers Computational time is decreased since

only a small amount of discretization lines are required in the computation avoiding the need to

solve large system of equations. ODE solvers like ODE45, ODE15s and ODE23 are preferred

because of their ability to handle stiff Equations.

In order to apply the Method of lines the following procedure is required

i. Partitioning the solution into two layers

ii. Discretization of the differential heat equation in one coordinate direction (space

variable)

iii. Transformation of the Partial differential equation to systems of ordinary

differential equation

iv. Applying the required boundary conditions

v. Then finally solving the systems of equations with the suitable ODE solver.

31

1.7.6 Comparison between finite difference method and method of lines

Even though finite difference could be used to solve the partial differential equation it has some

weakness as compared with the robust method of lines. The finite difference method has two

main approaches that is the finite explicit and the implicit methods. For the finite explicit the

major weakness is the constraint with the time step which should always be less than a half for

stability to be achieved.

Another problem is the error introduced in the time domain as a result of the finite difference

approximation which unlike the method of lines is left continuous and therefore has no error

introduced from the time domain. The finite explicit also requires a very large number of

iterations thereby increasing significantly the calculation time.

Regarding the implicit method which has no stability issue it still encounters errors as a result of

the finite difference approximation at both domains time and space. If the time steps are too large

it may become less accurate unlike the method of lines which retains its accuracy since there are

no time steps involved in its implementation Again the solution requires more computational

effort and storage in order to perform the solutions since the nodes are solved simultaneously

32

1.8 Scope of study

In previous work done by Kelkar et al. [26] to model the hydrate dissociation rate, a

mathematical model using Cartesian coordinates was employed that was based on Fourier’s heat

transfer equation. The resultant systems of equations were solved analytically. Peters et al. [1]

followed up on the previous work using cylindrical coordinates which was also based on Fourier

heat equation. The resultant systems of equations were solved using finite difference

approximation on both the time and space domain. He finally compared his model predictions

with the experiment he conducted using ice to produce the methane hydrate and subsequently

dissociated the plug by depressurization. The model prediction was within 5% of the

experimental data.

In this work, hydrate plug which is a problem in the oil and gas pipelines is simulated to find the

regimes where intrinsic kinetics and heat transfer is predominating at various range of

temperatures and pressures. The use of mathematical model which employs the numerical

method of lines was applied in the study of hydrate dissociation in this thesis. The mathematical

model based on heat conduction only by Fourier’s heat equation in cylindrical coordinates is

used. The heat model was used to determine the temperature profile with time. It was also used

to estimate the location of the interface with time. Then the kinetic model was used to investigate

regimes where the dissociation rate was heat controlled and also when it was intrinsic kinetic

controlled.

33

The Chapter 1 gives insight into hydrates in general and its occurrences and its thermodynamics

and kinetics. It concludes with a literature review of kinetics models by different authors and

methods. Then the computational procedure using numerical method of lines is applied to

discretize Fourier’s heat equation in cylindrical coordinates. The heat equation model developed

would be use to predict the dissociation rates for both structure I and structure II hydrates from

experimental data. Sensitivity analysis would be conducted to determine regimes where

dissociation rates are heat transfer controlled or intrinsic kinetics controlled. Chapter 2 deals with

application of method of lines and its correct implementation in deriving the heat equation in

cylindrical coordinates and also the kinetic equation for cylindrical coordinates. The Chapter 3

presents results of the numerical simulation and finally Chapter 4 discusses the conclusions and

recommendations.

34

CHAPTER 2 MODELLING OF HYDRATE PLUG DISSOCIATION RATE

2.1 Hydrate dissociation by double-sided depressurization

The dissociation of hydrate plug could be achieved by several methods. Among the methods,

includes, electrical heating, direct heating methods, chemical treatment and depressurization. In

this simulation study depressurization was used. Depressurization method involves a gradual

reduction of pressure in the pipewall without an external heat supply in order to destabilize the

thermodynamic equilibrium condition of the gas hydrate plug in the pipe .When the pressure is

reduced, sensible heat is supplied from the pipewall surroundings to both the hydrate interface

and vapour phase.

The dissociation of the hydrate plugs start as the heat move radially towards the hydrate plug

whereas the temperature at the pipewall is constant. The evolved gas is assumed to be

immediately removed from the surface of the dissociating plug and hence has negligible effect

on dissociation rate. The porosity of the hydrate is assumed to be uniform throughout the hydrate

plug. The dissociation temperature is also assumed uniform. The radial dissociation is the

dominant process rather than the axial dissociation as shown in Figure 13

Figure 13: Radial dissociation of hydrate plug Reproduced from Peters et al. [1]

35

Figure 14: Hydrate plug dissociation schematic Reproduced from Hong et al. [68, p. 1851]

Figure 14 illustrates a hydrate inner core which is surrounded by water layer next to the pipewall.

The temperature profiles are determined according to the Fourier’s law of heat conduction in

radial coordinates.

For water-hydrate phase only

…………………………………………………………………...........1

The boundary conditions in the system are given below;

36

At the pipewall

Tw=To r=ro, t>0……………………………………………………………………….…...2

At the interface

Tw=Td r=s1, t>0..............................................................................................................…...3

The Stefan condition

-

r=s1 t>0…………….….………………………..................4

The Stefan condition in equation 4 is obtain by the conservation of energy across the inter phase.

The interface velocity is determined by the Stefan [43] condition which is the thermal

conductivity, kw divided by the density of dissociating medium multiplied by the latent heat.

Therefore for a hydrate plug the dissociation rate,

is obtained by multiplying the temperature

gradient

by the Stefan condition [43] which will be

where ρH and γH are the density

of heat of dissociation of hydrate respectively. The hydrate porosity is given by θ which is

always less than one but greater than zero.

The boundary condition (2) indicates a constant temperature at the pipewall. The boundary

condition (3) also indicates the constant temperature at the water hydrate interface. Then the

boundary condition (4) also known as the Stefan condition shows that the heat conducted

through the water layer is equivalent to the heat required to dissociate hydrate plug.

There is no analytical solution to this system partial differential equation therefore it may be

solved by a numerical method.

Assuming the dissociation of a hydrate plug in a cylindrical region 0≤R≤ro is initially at a

constant pipewall temperature To, that is above the hydrate formation temperature at a given

pressure, P. When the pressure of the hydrate is reduced gradually until the temperature rises

37

above the hydrate equilibrium temperature, dissociation of the hydrate may start. However, if the

equilibrium temperature is below the ice point, ice will form around the hydrate as it dissociates

thus there would be three phases in the system (water-ice-hydrate). On the other hand if the

hydrate dissociation temperature is above ice point then only two phases would be present that is

water and hydrate.

When the pipeline is depressurized the hydrate dissociation temperature (Td) becomes less than

the pipewall temperature To. Thus, heat moves from the external pipe and travels radially into the

system to dissociate the hydrate plug. The hydrate plug begin to reduce in size as it detaches

itself radially from the pipe wall. The modeling of hydrate plugs was based on heat transfer by

conduction only.

The double-sided depressurization of the hydrate plug in the pipewall was considered as the

method for dissociation. The hydrate plug was assumed to undergo dissociation towards the

center of the pipe. As the hydrate plug decomposed, it became surrounded by stationary water

phase which transmitted heat to the dissociating front of the hydrate plug. Since the dissociation

temperatures, Td (273.25K) was greater than the ice point the hydrate system had only one

moving boundary.

38

2.2 Derivation of the moving boundary equation with the Kinetic term

In order to investigate the kinetics of the hydrate plug dissociation, the heat transfer model was

coupled with the kinetics. The coupling of intrinsic hydrate dissociation kinetics with heat

transfer rate is done in order to investigate the influence of kinetics in the overall rate of

dissociation. The approach was to vary both the system pressures and temperature independently

between a certain ranges in the simulation to observe those effects.

Clarke and Bishnoi [69] presented the total hydrate decomposition as the sum total rate of

evolution of each hydrate forming compound as follows;

……………………………………………. 5

NC is total number of hydrate-forming gases and Kd,j the intrinsic rate constant of the gas

hydrate dissociation rate for component j. Giraldo et al. [70]simplified the equation eliminating

the need for the intrinsic rate constant of the other hydrate forming gases apart from methane.

This was accomplished by relating the hydrate stoichiometric coefficients of the hydrate forming

compounds making the methane as the reference component as shown in equations below;

1M1 2M2 3M3…….. NC WH20……………………………………..........................6

where w1M1 w2M2, w3M3, and wNCw represent the stoichiometric coefficients and molar

masses of M1, M2, M3 and NCw respectively.

……………………………………………………………….......................... 7

39

Plugging equation 7 into 5 yield

........................................................................................................................................... 8

Sum of the relative stoichiometric ratios, W is given as;

Where W=

.......................................................................................................9

Hence the resulting dissociation equation is given as

………………………………………………………………...10

………………………………………………….. 11

Therefore the molar hydrate dissociation rate is given as;

……………………………………………………………........ …12

Where W (sum total of the relative stoichiometric ratios) is given by;

W=

……………………………………………………………………………… 13

And Kd the intrinsic dissociation constant

………………………………………………………………………......... 14

From the mass balance around the hydrate surface the molar rate of hydrate dissociation to the

plug thickness is given as;

…………………………………………………………………………...… .15

40

where Ageo is the geometric area of the hydrate plug and s is the interface location of the hydrate

plug,

The dimensionless surface roughness factor is given as;

Where

……………………………………………………………………........ ………16

Hence the kinetic model methane hydrate dissociation is used from Kim et al [40]

…………………………………………………………............17

When the heat transfer rate is combined with the kinetic equation we would be able to derive the

moving boundary system with the kinetic term.

Again applying the Stefan condition in equation 4

………………………………………………………….......18

Plugging (17) into (18) yields the equation 19 below

……………………………………19

Discretizing equation 19 gives the equation below

-Kw(

-

…………20

41

2.3 Derivation of moving boundary of the heat transfer model using method of

lines in radial coordinates

The partial differential equation (1) and its boundary conditions is solved numerically by the

method of lines. This numerical technique discretizes the space coordinates by finite difference

method while it leaves the time derivative continuous. This hybrid numerical methodology

transforms the partial differential equation to a system of ordinary differential equations which is

then solved using an appropriate ODE solver algorithm. The Fourier heat conduction equation in

cylindrical coordinates may be expressed by the following;

For moderate temperature range at constant pressure, dh (enthalpy change) can be replaced by

(temperature change.), dT.

dh=cpdT………………………………………………………………………………………... .21

……………………………………………………………………………... 22

The Stefan condition for a constant temperature at the hydrate – water interphase is given as;

……………………………………………………………………………………23

where L is latent heat of fusion of the solid phase and is the thermal diffusitiviyy of the

dissociated phase.

42

Figure 15: Discretization scheme with fictitious enthalpy values, Reproduced from Chun et al [71]

Figure 16: Schematic Linear enthalpy distribution for one-dimensional grid system, Reproduced from Chun et al [71]

43

2.4 Derivation of the heat equation by method of lines in the cylindrical

coordinates

In order to use the radial equation for the Fourier heat equation in equation (22) and the Stefan

condition in equation (23), Chun et al [71]approach are employed as shown in Figures 15 and 16

The linear enthalpy distribution for the one-dimensional radial equation in equation (22)

is discretized as shown in equation (24),with as fictitious nodal enthalpy

……………………………………………………....24

The finite difference approximation of equation (6) along the spacial coordinates is given below;

= -

………………………………………………………………….…………25

Solving for the fictitious nodes;

Since

= -

…………………………………………………………………………….26

setting hm to zero,

……………………………………………………………………………27

Plugging 10 into 7

…………………..................28

44

Simplifying equation (28)

………………………………………………..29

From equation 21 the enthalpy change dh can be represented by temperature change, dT the

equation 29 thus can be expressed in terms of temperature as below;

……………………………………………30

Then the discrete form in equation 30 can be expressed in the derivative form in the equation

below

…………………………………………………… 31

The equation (31) would be required to solve in cylindrical coordinates as was similarly

implemented by Campo and Ulises [72] in Cartesian coordinates by the application method of

lines in predicting the phase boundary velocity (dissociation rate) and temperature history.

45

1 2 3 i-1 i i+1 N-1 N

interface

∆r δ∆r (1-δ)∆r

Figure 17: Discretization Scheme at the interface, Reproduced from Lacoa et al. [66]

Problem Description

Using the discretization method obtained from [71, 73].The spatial domain is divided into N-1

uniform grid cells. The interface is assumed to be located between the nodes i and i+1 as

illustrated in Figure 17.The present model will be implementing the method of lines in

cylindrical coordinates. The spatial domain is divided into N-1 constant step size. Since the

interface is found between the lines i and i+1 as illustrated in Figure17, the line temperatures, Tk,

are determined from the systems of first order differential equations. Assuming constant thermal

properties, the mathematical formulation of the Stefan problem start with a one dimensional heat

46

conduction equation. The hydrate begins to dissociate from the pipewall r=0 at a constant

pipewall temperature at 277.15K.

The hydrate-water interface moves in a positive R-direction towards the center of the pipeline.

Hence at any time t, the temperature distribution and the radial position of r1 (t) could be

determined.

For the initial condition

T(r, t=0) = Ti……………………………………………………………………………………32

With the boundary conditions

T(r=Rw,t)=277.15K…………………………………………………………………………….33

Where the temperature at the pipewall is To and the temperature at the interface are given as

follows;

T(r=s(t),t)=273.25K…………………………………………………………………………….34

After applying the first boundary condition r=Rw, dissociation of the hydrate begins at this

location. The water hydrate interface moves inwards into the center of the plug in a positive R-

direction; r>0 to reach the moving boundary condition at R(t)

The governing heat conduction equation cannot be solved at the phase change interface and also

R(t) is not known a priori .Thus to overcome this problem the Stefan equation is introduced

which brings a closure to the set of equations. The Stefan condition indicates that the front

velocity (i.e. the dissociation rate by conduction) is proportional to the jump in heat flux across

the front. Generally, pure substances change phases isothermally and there is latent heat

associated with phase change (dissociation of hydrate). The latent heat of a given phase is the

47

heat liberated or absorbed when a unit mass undergoes that phase completely and isothermally.

The latent heat has dimensions of (energy/mass) which is given in units of joule/gram.

The Stefan condition is a physical constraint which comes from the conservation of energy at the

water-hydrate interface. Thus the local interface velocity (i.e. dissociation rate) depends upon the

flux discontinuity. The non-linear condition from equation 4 with initial condition at S(0)=0.The

density ,the thermal diffusivity of the water phase and heat of dissociation, of the hydrate is

provided in Table 8.

At r=0; the first derivative radial group becomes undefined in equation (1), therefore in order to

overcome this difficulty L’Hȏpital rule is applied by taking limit of the ratio of the derivatives as

r→0

…………………………………………………………………………………... 35

Thus at r=0 the equation below is obtained

……………………………………………………………………...36

For r>0 for K=2...i-1

………………………………………………………...37

While at i (i.e. interface), it is given by (from equation 31)

…………………………………………..…..38

Where the index i indicate

i=max (Tm K=1…N-1…………………………………………………………… …...39

48

is the dimensionless distance from line i to the interface from Figure 16

………………………………………………………………………………...40

Where 0 < …………………………………………………………………………….41

……at r=s (t)………………………………………………………………..42

There is singularity that arises from the hydrate-water interface as it gets extremely close to the

grid line. According to Verma et al [44] the temperature gradient at the interface at r=s(t) could

be evaluated .using. Taylor’s series expansion and taking only the first two terms .Therefore the

temperature gradient can be expressed as

……………………………………………………………..43

Hence plugging equation 24 into 23 yields

……………………………………………...44

Now in order to determine the temperature profile and the interface velocity (i.e. dissociation

rate), equations (37, 38 and 44) are applied respectively.

Therefore the systems of ODEs in equations (36), (37), (38) and (44) are solved using ODE15s

solver. Generally ODE assumes smoothness in the solution procedure but since there is a

discontinuity at the interface which would subsequently lead to structural changes. MATLAB

2016T and higher is equipped with event location which is able to identify the point of this

discontinuity to enable the integration to stop momentarily so it could be re-initialized for the

integration to start again.

49

The equilibrium fugacity is found using the Van der Waals Platteuw [74] theory. The fugacities

(fugacity at the interface) and (equilibrium fugacity) were determined by application of the

Trebble-Bishnoi Equation of state [75]. The activation energy, E and Universal gas constant, R

was obtained from Giraldo et al. [69] and geometric ratio, , was assumed to be unity for base

case. For the methane hydrate the stoichiometric factor, W was taken as unity. The

stoichiometric factors , W of the hydrate mixture gas mixtures with varying amount of mole

fractions of the methane and ethane were determined and presented in Table 4.

The porosity was assumed to be 0.5 which was a reasonable estimation since Lysne [23] showed

that porosities of plugs were ranged from 13% to 83%. For hydrate mixtures the calculated W

values was greater than unity. The parameter surface ratio is determined from Mullin et al [76].

They described the parameter as the ratio of the surface of the sphere having the same volume

as the particle to the apparent estimated surface area of the particle.

50

CHAPTER 3-RESULTS AND DISCUSSION

3.1: Simulation Results for Temperature Profile in the Pipeline

The major three driving forces hydrate dissociation mechanism are namely, heat transfer, mass

transfer and intrinsic kinetics. The Fourier heat equation for radial coordinates was applied to

model the process. The discretized equation 36, 37 generated the temperature profiles in the

radial coordinates.

The temperature profile illustrates a general increase in the water phase temperature at

decreasing radial positions from the pipewall. When the hydrate dissociate at the interface, it

absorbs heat that is transmitted from the pipewall though the water phase or water layer around

the hydrate phase. Temperature gradient was established between the pipewall and the hydrate-

water interface at various radial positions. The temperature profile in the water phase depends on

the rate of heat consumed at the interface and the thermal properties of both the water and the

dissociating hydrate phase.

Table 3: Table showing the number of iterations for the method of lines using different ODE solvers

ODE15s 118

ODE 23s 604

ODE45 7477

The ODE 15s was used in our simulations it had the lowest iterations to achieve convergence.

51

Figure 18: Plot of temperature profiles at different radial positions

52

Figure 19: Plot of Temperature profiles of both pure methane (C1) and mixture of methane/ethane (C1/C2) hydrate

composition at different radial positions

53

Figure 20:3 D Plot of Temperature Profile of methane/ethane mixture at different radial positions

54

Figure 21: 3 D Plot of Temperature Profile of pure methane at different radial positions

55

Figure 22: Plot of Temperature Profiles with pure methane and mixture at different radial positions using kinetic model

56

Figure 23: 3D plot of pure methane at different radial position using the kinetic model

57

3.2 Decomposition of Hydrate plug kinetics

. The model presented here is to describe the depressurization mechanism in the pipe by coupling

the intrinsic kinetics of the hydrate dissociation with the heat transfer. The model tracks the

movement of the hydrate interface which involves only one moving boundary problem that is

water and hydrate only. The model also reveals the range of the pressures where the heat

transfers resistances is dominating as well as the range where the intrinsic kinetics of the hydrate

dissociation is predominant or kinetics controlled.

The model is initially applied to a finite one-dimensional cylindrical pipeline as presented in

Figure 17. The heat model equation 38 is then matched with the experimental data. The heat

transfer model is then finally extended to predict the hydrate dissociation rate and matched with

the data from Peters and Sloan [77] on the hydrate dissociation rate for both the hydrate plugs for

structures I and II. The heat transfer model developed in this work was validated by matching the

simulated results with the data from Peters and Sloan [77].

The hydrate plug porosities were assumed to be 0.5.The model predicted the position of the

dissociating front of the hydrate plug as a function of time for each of the experimental data. The

latent heat of dissociation for structure I is 460.24kJ/kg, while that of structure II hydrates is

640.15kJ/kg [1] . There was good in agreement between the model prediction for structure II and

the data with an AAD of less than 5%.

58

The possible causes of the deviation from the data could be attributed to the experimental error,

equipment error or set-up and environmental conditions. The dissociation rate for structure I was

faster than structure II. The faster dissociation rate of structure I was probably due to its lower

latent heat of dissociation. The radial dissociation of the hydrate was assumed to be in the center

of the pipeline during the dissociation while the remaining space was occupied by the water.

However, buoyancy effects as well as the presence of hydrocarbon phase on the fluid thermal

diffusivity of the aqueous phase might have influenced the heat transfer rate.

The surrounding fluid composition and the liquid volume fraction for an industrial hydrate plug

will depend on the flowline geometry and plug location. The porosity is not known a priori on

the industrial scale a value of 0.5 was assumed because it was the average value which is a fair

representation of most hydrate plugs. The experimental data that was used to validate the heat

model had the initial equilibrium pressure of 9.8MPa with 7.8MPa as the initial system pressure

at a temperature of 273.25K.

Further simulations using 5.4MPa and 3.8MPa at the same temperature 273.25K was done to

find out which pressure regimes the dissociations rates were kinetically controlled. The equation

37 was applied to predict the dissociation rate in a heat transfer limited system. The validation of

the present model was used to fit the experimental data of structures I and II of gas hydrate plug

dissociation data of Peters and Sloan. It was found that the percentage average absolute deviation

(% AAD) of the predicted hydrate dissociation rate for structure1 were less than 4% from the

experimental data [77] as illustrated in Figure 24.

59

Table 4: Table of calculated stoichiometric values, W of hydrate mixtures (CH4+C2H6)

Components W Type of Hydrate

structure

Pure methane 1.00 Structure I

65%CH4+35%C2H6 2.66 Structure II

85%CH4+15%C2H6 2.85 Structure II

95%CH4+5%C2H6 2.99 Structure II

Table 5: Simulation Parameters of the hydrate plug

Radius of hydrate plug, ro 50cm

Dissociation temperature, Td 273.18K

Pipewall Temperature, To 277.15K

Porosity, θ 0.5 unitless

Geometric ratio for base case, Ψ 1.0 unitless

Stoichiometric coefficient for gas mixture 2.66 unitless

60

Table 6: Table of activation and intrinsic kinetics constants for methane and ethane hydrates

source [42]

Species

∆E(kJ/mol)

Kdo (mol m

-2 Pa

-1s

-1)

CH4 in s1 81.0 3.60*104

CH4 in s11 77.3 8.06*103

C2H6 in s1 and s11 104.0 2.56*108

61

Figure 24: Plot of structure I hydrate dissociation rate compared with the experimental data [77]

0

10000

20000

30000

40000

50000

60000

0 1000 2000 3000 4000 5000 6000

Co

mm

ula

tive

vo

l/L

Time/s

comparison of sI data with heat transfer model Model Data

62

Figure 25: Plot of dissociation rate heat model for structure I with experimental data of Peters et al. [77]

The accuracy of the prediction of the hydrate dissociation rate depends on parameters which are

needed by the heat transfer model based on the assumptions associated with the model. There is

an uncertainty of the value of the porosity in the model. A value of 0.5 has been assumed .The

porosity value depends on the original characteristics of both the hydrocarbon and the source of

geological formation of the hydrocarbon. In the present work both structures I and II hydrates

used the same assumed porosity value of 0.5. The Figures 24 and 25 shows that the model agrees

fairly well with the experimental data for both structures I and II.

0

10000

20000

30000

40000

50000

60000

0 1000 2000 3000 4000 5000 6000

cum

mu

lati

ve v

ol /

L

Time/s

comparison of sII data with heat transfer model Model Data

63

At the initial stage there is a little under prediction of the model values with the experimental

data probably due to the intrinsic kinetics but as the dissociation progresses to the latter stages

there is a significant improvement of the model prediction which eventually gives a very good

match with the data. This satisfactorily good agreement between the model and data suggest that

the assumptions stated in our model were quite valid

64

3.3 Sensitivity analysis of the Model

The model was simulated to investigate pressure regimes where intrinsic kinetics became more

predominant. Therefore a the pressures were varied in the simulation to identify which pressure

regimes indicated where intrinsic kinetics were pre-dominanting.and also where it has little or no

influence. The following results show the trend analysis of the temperature, pressure and the

geometric ratios.

Figure 26: Plot of heat transfer model (HT) compared with kinetic model at different pressures. at a geometric ratio of

one

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3 4 5 6 7 8 9 10

no

of

mo

les

of

gas

evo

lve

d/n

Time/hours

Plot of no of moles against time at different Pressures(T=273.15K,Ψ=1)

Ht 3.8Mpa

HT 5.8MPa

Model 3.8MPa

HT 7.8MPa

Model 5.8 MPa

Model 7.8 Mpa

65

The sensitivity analysis illustrates that by changing the pressure from 7.8MPa to 3.8MPa the

dissociation rate moves from heat controlled to a regime where intrinsic kinetics begins to

predominate the process. As can be seen from Figure 26 the percentage deviation between the

heat transfer and the kinetic model is less than one percentage. However reducing the pressure to

5.8MPa and finally to 3.8MPa shows a sharp deviation from the heat transfer model. The

significant deviation indicates that the intrinsic kinetics becomes more prominent when the

pressure is decreased to 3.8MPa and 5.8MPa respectively.

66

Figure 27: Plot of heat transfer (HT) model with present model (Kinetic) at different temperatures

The second parameter that was investigated was the temperature. As can be seen in Figure 27 the

temperature at 273.15K showed that the dissociation rate was only heat transfer controlled at the

pressure of 7.8MPa .The AAD was less than 1% .However as the temperature was increased to

275.15K and eventually to 277.15K there was significant deviation from the model to the heat

transfer model. As the temperatures were raised intrinsic kinetics became more significant, this

could be attributed to the rate constant which is dependent on temperature. Therefore the

increased temperature raises the value of the rate constant thus the overall rate of dissociation is

ultimately increased.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6 8 10

no

of

mo

les

gas

evo

lve

d/n

Time/hours

Plot of no of moles against time(P=7.8Mpa,Ψ=1)

273.15K HT

275.15K HT

277.15K HT

273.15K model

275.15K model

277.15K model

67

Figure 28: Plot of hydrate kinetics using different geometric parameters

The last parameter that was investigated was the geometric factor. As can be seen from Figure 28

the model was not very sensitive to the geometric factor. There was not a significant difference

between the different geometric factors indicating that the factor was very negligible.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3 4 5 6 7 8 9 10

no

of

mo

leso

f g

as e

volv

ed

/n

Time/hours

Plot of different different geometric values versus time for the present model

Ψ=1---Ψ =2---Ψ=4

68

CHAPTER 4: CONCLUSIONS AND RECOMMENDATIONS

4.1 Conclusions

A moving boundary heat conduction model was formulated to describe the decomposition of a

pipeline hydrate plug. The model includes, in one of the boundary conditions, a term that can

describe the rate of gas hydrate decomposition for hydrates that have been formed in the

presence of a gas mixture. At the time of writing no other modeling attempt had included a term

to explicitly account for the composition of the gas phase, in the heat equation. The dissociation

of a hydrate plug was studied using a very powerful method of lines to discretize the Fourier heat

equation in radial coordinates. The simulation results showed that the hydrate dissociation raw

data matched well the heat transfer model with a deviation of 4% and 5% for structure I and

structure II hydrates plug respectively. It was then coupled with the intrinsic kinetics with the

heat transfer rates.

The simulation results indicated that the rate of hydrate dissociation was sensitive to several

factors which include system pressure, temperature and roughness of the surface of the hydrate.

It was found that by changing the pressure from 7.8MPa to 3.8MPa we could move from a heat

controlled regime to a regime where both heat transfer and intrinsic kinetics have profound effect

on the global rate of dissociation. Also changing the temperature from 273.15K to 277.15K the

system moved from heat controlled to kinetic controlled. The numerical method of lines was the

technique that was used to discretize the heat equations. It was then coupled with the kinetics to

develop the Kinetic model.

69

4.2 Recommendations

From the simulation results of this study, the following suggestions and recommendations must

be considered for future studies:

1) Experiments should be performed to track the decomposition of a hydrate plug in which

the hydrate was formed in the presence of a gas mixture.

2) The model could also be investigated for three phase behavoiur where ice could be

present with the hydrate in the pipe.

3) The model derived in this study could be used as a starting point of a gas hydrate plug

that is subject to a single sided depressurization.

70

APPENDIX

Table 7: Simulation Parameters source [27]

Activation Energy, ∆E 81.0 kJ/mol

Universal gas constant, R 8.3146 JK-1

mol-1

Porosity, θ 0.5

Intrinsic rate constant, Ko(CH4) 3.60*104 mol m

-2 Pa

-1 s

-1

Geometric ratio of hydrate plug, 1.0

Thermal conductivity of water ,kw 1.31*10-3

cm2/s

Thermal conductivity of ice, KI 1.31*10-2

cm2/s

Thermal diffusivity of ice ,αI 1.0007*10-3

m2/s

Thermal diffusivity of water ,αw 1.486*10-7

m2/s

Density of hydrate, 914 kg/m3

Dissociation temperature, Td 0.3oC

Density of ice, kI 917kg/m3

71

Table 8: Hydrate dissociation models, Reproduced from Sloan and Koh [10]

Model Heat transfer Fluid flow Kinetics Solution

Method Conduction Convection Gas Water

Holder and

Angert(1982)

X X Numerical

Burshears et

al(1986)

X X X Numerical

Jamuladin et al

(1989)

X . X Numerical

Selim and

Sloan(1989)

.......X .....X ......X Analytical

Yousif and

Sloan(1991)

...... X X ......X Numerical

Makogon(1997) X X X ......X Analytical

Tsypkin(2000) .......X ......X .....X ......X Analytical

Masuda et al

(2002)

X X .....X ......X .....X Numerical

Moridi et

al(2002)

......X .....X X .......X X Numerical

Pooladi-

Darvish et al

(2003)

X .....X X .......X X

72

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